A square matrix is called a diagonal matrix if it has 0 on all positions except the main diagonal. For example: Identity matrices of all orders are diagonal matrices. Symmetric Matrix: A square matrix where the
True or False: If A and B are symmetric n by n matrices, then A + B is also symmetric. An m \times n matrix has m columns and n rows. True False Determine whether the statement is true or false. Justify your answer. T...
Since A is symmetric it is diagonalizable with respect to an orthonormal basis, and the corresponding eigenvalues are real, and so can be ordered as follows: The first eigenvalue of a d-regular graph satisfies and has the all-ones vector as the associated eigenvector. Let . A graph G is...
One possible explanation for this phenomenon is that two diagonalizable matrices have the same spectrum if they are similar via an arbitrary similarity transformation rather than just via a permutation transformation, and this suggests considering features of a matrix that are invariant under permutation ...
(for a real symmetric matrix) is a bitsubtle,and we will come back to it later sections.The second important property of real symmetric matrices is that they are alwaysdiagonalizable,that is,there is always a basis for Rnconsisting of eigenvectors forthe matrix.111112 III.APPLICATIONSExampl...
Why are symmetric matrices diagonalizable? Suppose ab = ac, where b and c are n by m matrices and a is invertible. Explain why b = c. Explain why this is not true if a is not invertible. Can a square matrix with two identical rows be invertible?
Why are projection matrices symmetric? Are positive definite matrices always symmetric? If a is a symmetric matrix, what can you say about the definiteness of a^2? When is a^2 positive definite? Are all positive Semidefinite matrices symmetric? Why are symmetric matrices diagonalizable? Consid...
Degenerate Matrix: A matrixAis said to be diagonalizable if there exists an invertible matrixPsuch thatP−1APis a diagonal matrix. A degenerate matrix is a matrix whose determinant is zero. Answer and Explanation:1 A matrix of ordern×nis said to be diagonalizable if it hasneigenvectors. ...
Matrix Properties: Consider a constantcand a matrix[A]m×n, the product ofcwith[A]m×ngivesc[A]m×n, that is, each and every entry is multiplied byc. For the addition or subtraction of matrices[A]and[B],m×nshould be the same. Here,m×nis equi...
Matrix consists of number that are arranged in rows and columns, they can be used to represent vectors. A coefficient matrix is a matrix which consists of the coefficient of x variables. Matrices can be manipulated in many ways including addition, multiplication, subtraction....